International Journal of Heat and Mass Transfer 112 (2017) 267–278

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Uncertainty quantification of thermal conductivities from equilibriummolecular dynamics simulations

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.04.0770017-9310/� 2017 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: School of Mechanical Engineering, Purdue University,West Lafayette, IN 47907, USA.

E-mail address: ruan@purdue.edu (X. Ruan).

Zuyuan Wang a,b, Salar Safarkhani a, Guang Lin a,c, Xiulin Ruan a,b,⇑a School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USAb The Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907, USAcDepartment of Mathematics, Purdue University, West Lafayette, IN 47907, USA

a r t i c l e i n f o

Article history:Received 14 December 2016Received in revised form 6 February 2017Accepted 17 April 2017Available online 4 May 2017

a b s t r a c t

Equilibrium molecular dynamics (EMD) simulations along with the Green-Kubo formula have beenwidely used to calculate lattice thermal conductivities. Previous studies, however, focused primarily onthe calculated thermal conductivities, with the uncertainty of the thermal conductivities remainingpoorly understood. In this paper, we study the quantification of the uncertainty by using solid argon, sil-icon, and germanium as model material systems, and examine the origin of the observed uncertainty. Wefind that the uncertainty increases with the upper limit of the correlation time, tcorre;UL, and decreaseswith the total simulation time, ttotal, whereas the velocity initialization seed, simulation domain size, tem-perature, and type of material have minimal effects. The relative uncertainties of the thermal conductiv-ities, rkx=kx;ave, for solid argon, silicon, and germanium under different simulation conditions all follow asimilar trend, which can be fit with a ‘‘universal” square-root relation, as rkx=kx;ave ¼ 2ðttotal=tcorre;ULÞ�0:5.We have also conducted statistical analysis of the EMD-predicted thermal conductivities and derived aformula that correlates the relative error bound (Q), confidence level (P), tcorre;UL; ttotal, and number of inde-pendent simulations (N). We recommend choosing tcorre;UL to be 5–10 times the effective phonon relax-ation time, seff , and choosing ttotal and N based on the desired relative error bound and confidencelevel. This study provides new insights into understanding the uncertainty of EMD-predicted thermalconductivities. It also provides a guideline for running EMD simulations to achieve a desired relative errorbound with a desired confidence level and for reporting EMD-predicted thermal conductivities.

� 2017 Elsevier Ltd. All rights reserved.

1. Introduction

Equilibrium molecular dynamics (EMD) simulation along withthe Green-Kubo formula is an effective way to calculate latticethermal conductivities [1–9]. In this method, the thermal conduc-tivity is related to the integration of the heat current autocorrela-tion function (HCACF), as [4]

kab ¼ V

kBT2

Z 1

0hJað0ÞJbðtcorreÞi dtcorre; ð1Þ

where kab is the abth component of the thermal conductivity tensor,V is the volume of the material system, kB is the Boltzmann con-stant, T is temperature, tcorre is the heat current autocorrelationtime, and Ja is the ath component of the full heat current vector J,which is typically computed as [10]

J ¼ 1V

Xi

vi�i þXi

Si � vi

!; ð2Þ

Here, vi; �i, and Si are the velocity, energy, and stress of atom i. InLAMMPS, [10] a widely used, open-source molecular dynamics sim-ulation package, the default heat current formula is based on Eq. (2)with the interatomic forces calculated from the per-atom stresses.Recently Fan et al. reported new heat current formulas for many-body potentials, but the different heat current formulas are shownto affect mainly low-dimensional materials [11]. In theory, the V,integration upper limit, and heat current autocorrelation time inEq. (1) should all approach infinity to calculate the lattice thermalconductivity of bulk materials. In real practice, however, the V ischosen to be of a finite size based on some domain size effect stud-ies, the integration is carried out up to a finite upper limit, which wedefine as the upper limit of the correlation time, tcorre;UL, and theheat current autocorrelation is calculated up to a finite duration,which we define as the total simulation time, ttotal. As a result, Eq.(1) becomes

268 Z. Wang et al. / International Journal of Heat and Mass Transfer 112 (2017) 267–278

kab ¼ V

kBT2

Z tcorre;UL

0hJað0ÞJbðtcorreÞijttotal dtcorre: ð3Þ

Although this method has been widely used to calculate the lat-tice thermal conductivity of many material systems, previous stud-ies focused primarily on the thermal conductivity values or theaverage values from multiple independent simulations, with theuncertainty of the predicted thermal conductivities remainingpoorly understood, as seen from the very limited investigationson it so far [12–14]. On the other hand, it is a common practiceto report both the values and uncertainties of the predicted ther-mal conductivities from EMD simulations. A typical way of doingit is to run each simulation for multiple times (usually 3–12) andthen calculate the average value as the thermal conductivity andthe standard deviation as the uncertainty (plotted as error bars).This practice, however, often lacks consistency (or a well-definedguideline) because the values and uncertainties could vary greatlydepending on how the simulations are conducted. In addition, itwas pointed out that the uncertainty of the thermal conductivityfrom EMD simulations is about 20%, [3] for which no explanationwas provided. Furthermore, when thermal conductivities fromEMD simulations are compared with those from other sources(e.g., experiments or other simulation methods), it is often con-cluded that the agreement is good if the error bars overlap, but lit-tle is known about the information carried by the error bars.

In this study, we conduct a systematic study on quantifying theuncertainty of thermal conductivities from EMD simulations. Weconsider solid argon, silicon, and germanium as model materialsystems, and study the effects of the velocity initialization seed,simulation domain size, upper limit of the correlation time(tcorre;UL), total simulation time (ttotal), temperature, and type ofmaterial. The results show that the uncertainty increases withtcorre;UL and decreases with ttotal, but the velocity initialization seed,simulation domain size, temperature, and type of material haveminimal effects on the relative uncertainty. By analyzing theresults of different materials under different simulation conditions,we have obtained a ‘‘universal” square-root relation for quantifyingthe relative uncertainty, rk=kave, as a function of ttotal=tcorre;UL. Wehave also obtained a formula that correlates the relative errorbound (Q), confidence level (P), tcorre;UL; ttotal, and number of inde-pendent simulations (N). This paper is organized as follows. Sec-tion 2 details the method used in this study, particularly theEMD simulations. Section 3 presents some results and discussionon the uncertainty of the EMD-predicted thermal conductivitiesof solid argon, silicon, and germanium. It also reports some resultson quantifying the general uncertainty of EMD-predicted thermalconductivities, choosing appropriate tcorre;UL and ttotal for EMD sim-ulations to achieve a desired relative error bound with a desiredconfidence level, and reporting EMD-predicted thermal conductiv-ities. Section 4 summarizes the main findings from this study.

2. Methodology

All the molecular dynamics simulations were conducted withthe LAMMPS package [10]. The material systems of solid argon, sil-icon, and germanium all have a face-centered cubic (FCC) structurewith nominal lattice constants (before the structures are relaxed)of 5.26, 5.43, and 5.66 Å, respectively. The interatomic interactionsare characterized with the Lennard-Jones potential [15] for solidargon and the Tersoff potential [16] for silicon and germanium.We considered a domain size of 6� 6� 6 unit cells (u.c.) for solidargon (except for the domain size effect studies) and 4� 4� 4 u.c.for silicon and germanium. Periodic boundary conditions wereapplied in x; y, and z directions. The time steps were chosen as4, 1, and 2 fs for solid argon, silicon, and germanium, respectively.Nosé-Hoover barostat and thermostat [17,18] were used to control

the pressure and temperature of the material systems. In all simu-lations, the material systems were first equilibrated in an NPT (con-stant number of atoms, pressure, and temperature) ensemblebefore they were switched to an NVE (constant number of atoms,volume, and energy) ensemble for data production. The tcorre;ULand ttotal values were chosen such that the predicted average ther-mal conductivities converged. We varied the tcorre;UL and ttotal over awide range to investigate their effects on the uncertainty of thepredicted thermal conductivities. Each simulation was run for100 times, which had independent initial velocity distributions. Itis an inherent assumption in this study that 100 independent sim-ulations provide a representative sample for the relevant statisticalanalysis. The thermal conductivities were calculated according toEq. (3). Since each individual EMD simulation can provide threethermal conductivity values (for the x; y, and z directions), thereare a total of 300 thermal conductivity values for each simulationcondition. We report the average and standard deviation of the300 values as the predicted thermal conductivity and its uncer-tainty, respectively. Because the three materials considered in thisstudy are all isotropic in the x; y, and z directions, the 100 kx, 100ky, and 100 kz values for each simulation condition could be equiv-alently treated as 300 kx values. As a results, the analysis in thisstudy essentially corresponds to the thermal conductivity along asingle direction. Alternatively, the thermal conductivities can befirst averaged over the x; y, and z directions and then the averageand standard deviation of the 100 values from the 100 simulationscalculated as the predicted thermal conductivity and its uncer-tainty, respectively. The average from these two methods will bethe same, but the standard deviation from the second method willbe statistically 1=

ffiffiffi3

ptimes that from the first method. Considering

the second method is restricted to isotropic materials, we adoptedthe first method to make our analysis more general.

3. Results and discussion

In this section, we present the results for the three materials —solid argon, silicon, and germanium. For solid argon, we show theeffects of the velocity initialization seed, simulation domain size,ttotal; tcorre;UL, and temperature on the EMD-predicted thermal con-ductivity and its uncertainty. For silicon and germanium, we focuson the effects of the ttotal and tcorre;UL. Based on the solid argon, sil-icon, and germanium results, we provide some consideration onquantifying of the general uncertainty of EMD-predicted thermalconductivities. We also show how to appropriately choosetcorre;UL; ttotal, and N for EMD simulations so that the predicted aver-age thermal conductivity achieves a desired relative error boundwith a desired confidence level and how to report EMD-predictedthermal conductivities.

3.1. Solid argon

In EMD simulations, independent simulations are usually con-ducted to reduce the statistical error, which can be realized byassigning different velocity initialization seeds. In LAMMPS, theonly requirement for a velocity initialization seed is that it be apositive integer [10]. To understand how the seeds affect the ther-mal conductivity predictions, we considered two schemes ofassigning the seeds, namely, uniform and random seeds. The uni-form seeds are described as 1000n, where n is the simulation ID(varying from 1 to 100), whereas the random seeds are randomnumbers (from 1000 to 100,000) generated with the rand functionof MATLAB. In Fig. 1(a), we show some typical HCACF profiles forsolid argon. It is seen that the normalized HCACF starts from one,decreases gradually to zero, and then fluctuates around zero. Typ-ically the correlation time, tcorre;UL, should be long enough so that

Fig. 1. (a) Typical heat current autocorrelation function (HCACF) profiles of solidargon, normalized by the initial HCACF values. (b) Variation of the EMD-predictedthermal conductivity of solid argon at 40 K with the simulation domain size. Eachdata point shows the results of 100 independent simulations (or 300 values). Thedata points for the uniform seeds are shifted slightly to the left for betterreadability. The black (dashed) and green (dotted) lines show the experimentalresult (0.560 W/m-K) by Touloukian et al. [19] and simulation result (0.574 W/m-K)by McGaughey and Kaviany, [5] respectively. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

Z. Wang et al. / International Journal of Heat and Mass Transfer 112 (2017) 267–278 269

the HCACF profiles cross zero for multiple times. The results inFig. 1(a) indicates that tcorre;UL ¼ 80 ps is sufficiently long for calcu-lating the thermal conductivity of solid argon at 40 K. The discrep-ancy of the HCACF profiles in x; y, and z directions could beattributed to the finite domain size, finite total simulation time,and statistical nature of molecular dynamics simulations. InFig. 1(b), we show the domain size effect of the EMD-predictedthermal conductivity of solid argon, including the results with boththe uniform and random seeds. Note that the data points for theuniform seeds are shifted slightly to the left for better readability.It is seen that the thermal conductivity increases with the increas-ing domain size and converges at a size around 6� 6� 6 u.c. Theresults with uniform and random seeds differ appreciably for smalldomain sizes, but they agree reasonably well for domain sizes lar-ger than 6� 6� 6 u.c. The relative uncertainty of the thermal con-ductivities is around 25%. We will show that this level ofuncertainty comes primarily from the choices of tcorre;UL and ttotal.Upon convergence, the average thermal conductivities with boththe uniform and random seeds agree well with the experimentalvalue (0.560 W/m-K) by Touloukian et al. [19] and the simulationresult (0.574W/m-K) by McGaughey and Kaviany [5].

In Fig. 2, we show the detailed thermal conductivity distribu-tions corresponding to the different simulation conditions. It isseen that the histogram distributions of the thermal conductivitiesunder different simulation conditions all agree reasonably wellwith the corresponding normal distribution curves. Comparing

the results in Fig. 2(a) and (d), we notice that at a domain size of2� 2� 2 u.c., tcorre;UL ¼ 80 ps, and ttotal ¼ 2:4 ns, significant discrep-ancies exist between the calculated average thermal conductivitywith the uniform seeds and that with the random seeds. As thesimulation domain size increases, the discrepancy decreases (seeFig. 2(b) and (e) or Fig. 2(c) and (f)), which could be attributed tothe more ergodic sampling of the phase space at a larger domainsize. The standard deviations of the thermal conductivity distribu-tions are comparable for domain sizes of 6� 6� 6 and10� 10� 10 u.c., suggesting a weak dependence of the standarddeviation on the domain size upon convergence. We also consid-ered the effects of ttotal and tcorre;UL. As shown in Fig. 2(g) and (h),the average and standard deviation of the thermal conductivitieswith the uniform and random seeds agree well at ttotal ¼ 76:8 ns,even at a small domain size of 2� 2� 2 u.c. This implies that thelimited sampling of the phase space due to the limited domain sizecould be compensated by using a long total simulation time. Com-paring the results in Fig. 2(b) and (i), we observe a similar averagethermal conductivity but a much flatter distribution (meaning alarger standard deviation) of the thermal conductivities, astcorre;UL increases from 80 to 160 ps.

After noticing the significant effects of ttotal and tcorre;UL, we con-ducted a more detailed study on their effects on the predictedaverage thermal conductivity and the uncertainty. Fig. 3 showsthe HCACF and thermal conductivity integration profiles for solidargon at 40 K, which correspond to a domain size of 6� 6� 6 u.c.and tcorre;UL ¼ 40 ps but different ttotal values. The value for tcorre;ULwas chosen based on some preliminary simulation results (seeFig. 1(a)). We will provide some guidance on choosing an appropri-ate tcorre;UL in Section 3.5.1. As ttotal increases, the HCACF profilesbecome smoother, and the difference between the HCACF profilesin different directions decreases. As a result, the distribution ofthe thermal conductivities becomes increasingly concentrated, asttotal increases. We point out that the commonly seen fluctuationsof the HCACF profiles around zero is non-intrinsic, because theycould be reduced and eventually eliminated by increasing the totalsimulation time. For the thermal conductivity distribution, whenttotal is very small, abnormal thermal conductivities (such as nega-tive or extremely large values) could appear; when ttotal is large, thethermal conductivities from the independent simulations form anarrow band, approaching a single value in the theoretical limit.Regarding the average thermal conductivity, all simulation condi-tions result in a similar value, which, to some extent, demonstratesthe equivalence of time-averaging and ensemble-averaging. [20].

We consider further the effects of ttotal and tcorre;UL on the EMD-predicted average thermal conductivity of solid argon and theuncertainty. Fig. 4(a) shows the variation of the average thermalconductivity with ttotal at three tcorre;UL values. When ttotal is short,the average thermal conductivity differs greatly from the experi-mental value. When ttotal is longer than a few nanoseconds, theaverage thermal conductivity becomes stable. The converged aver-age thermal conductivity generally increases with tcorre;UL, with thevalue in good agreement with the experimental one for ttotal longerthan 40 ps. This confirms the appropriateness of the choice oftcorre;UL for the results shown in Fig. 3. Fig. 4(b) shows the variationof the thermal conductivity distribution with tcorre. As tcorreincreases, the distribution expands, indicating a larger uncertaintyof the predicted thermal conductivities. The insets show the his-togram distributions of the thermal conductivities correspondingto tcorre;UL ¼ 40, 60, and 80 ps, respectively, in comparison withthe normal distribution curves. Fig. 4(c) shows the variation ofthe thermal conductivity distribution with ttotal. As ttotal increases,the average remains nearly constant, but the error bars becomesmaller, indicating a smaller uncertainty of the predicted thermalconductivities. The insets show the histogram distributions of the

Fig. 2. Histogram distributions of thermal conductivities of solid argon from 100 independent simulations (or 300 values) with uniform or random velocity initializationseeds. The simulation conditions are indicated in the subfigures. The bell-shaped curves in the subfigures represent normal distribution curves with the same average thermalconductivity and uncertainty.

270 Z. Wang et al. / International Journal of Heat and Mass Transfer 112 (2017) 267–278

thermal conductivities corresponding to different ttotal values, incomparison with the normal distribution curves. As ttotal increases,the histogram distribution becomes more and more peaked, con-firming the decreasing uncertainty of the predicted thermalconductivities.

Having seen the negligible effects of ttotal and tcorre;UL on theEMD-predicted average thermal conductivity of solid argon andtheir qualitative effects on the uncertainty, we move on to quanti-tatively understand the effects of ttotal and tcorre;UL on the uncer-tainty of the EMD-predicted thermal conductivity of solid argon.In Fig. 5(a), we show the variation of the standard deviation ofthe thermal conductivities with tcorre;UL. For all ttotal values, rk

increases as tcorre;UL increases. We fit the data with a power lawrelation as rk � tacorre;UL, and the resulted a varied from 0.48 to0.52. In Fig. 5(b), we show the variation of the standard deviationof the thermal conductivities with ttotal. For all tcorre;UL values, rk

decreases as ttotal increases. We fit the data with a power law rela-tion as rk � t�b

corre;UL, and the resulted b again varied from 0.48 to0.52. These results suggests a square-root relation as

rk=kave � ðttotal=tcorre;ULÞ�0:5, where the average thermal conductiv-

ity, kave, is used to nondimensionalize rk. After plotting rk=kavewith ttotal=tcorre;UL, we realize that the square-root relation indeedholds, especially for ttotal=tcorre;UL in the range from 5 to 2000, asseen in Fig. 5(d). To confirm that the relation holds true for otherdomain sizes and temperatures, we conducted some additionalEMD simulations with domain sizes of 10� 10� 10 and20� 20� 20 u.c. and at another temperature 80 K. Surprisingly,all data follows a similar trend, as seen in Fig. 5(d). We fit all thedata with the square-root relation, and the result shows

rk=kave ¼ 2:025ðttotal=tcorre;ULÞ�0:5. As a test of this relation, we con-sider the EMD results in Fig. 1(b). It turns out that the predictedrelative uncertainty (rk=kave ¼ 37%) by the square-root relation isin excellent agreement with the actual relative uncertainty(around 36%). In the following sections, we will show that this kindof square-root relation is not limited to solid argon.

3.2. Silicon

Considering the significant effects of ttotal and tcorre;UL on theuncertainty of EMD-predicted thermal conductivities of solid

Fig. 3. (a)–(d) Typical HCACF profiles of solid argon with a domain size of 6� 6� 6 u.c. at 40 K. The simulations correspond to tcorre;UL ¼ 40 ps but different ttotal values. In (d)the results based on single- and double-exponential fitting are included to show how the effective phonon relaxation time, seff , could be obtained. (e)–(h) Distributions ofthermal conductivity integration profiles of solid argon. The simulation conditions are indicated in the subfigures. The red (thin) curves represent the thermal conductivitiesfrom 100 independent simulations (including a total of 300 curves). The blue (thick) curves represent the corresponding average thermal conductivities. (For interpretation ofthe references to color in this figure legend, the reader is referred to the web version of this article.)

Z. Wang et al. / International Journal of Heat and Mass Transfer 112 (2017) 267–278 271

argon, we focus on the effects of these two parameters in the studyon the silicon material system. In Fig. 6, we show the HCACF andthermal conductivity integration profiles for silicon at 500 K, whichcorrespond to a domain size of 4� 4� 4 u.c. and tcorre;UL ¼ 400 psbut different ttotal values. Similar to the solid argon results, thevalue of tcorre;UL was chosen based on some preliminary simulationresults. As ttotal increases, the HCACF profiles become smoother,and the difference between the HCACF profiles in different direc-tions decreases. As a result, the distribution of the thermal conduc-tivities becomes more concentrated, as ttotal increases. Comparedwith the results for solid argon (see Fig. 3), the fluctuations ofthe HCACF profiles for silicon is much larger, which could beattributed to the optical phonons as there are two basis atoms ina primitive unit cell of silicon. Still, the fluctuations are considerednon-intrinsic, because they could be reduced and eventuallyeliminated by increasing the total simulation time. For the thermalconductivity distribution, when ttotal is very small, abnormal ther-mal conductivities (such as negative or extremely large values)could appear; when ttotal is large, the thermal conductivities fromthe independent simulations form a narrow band, approaching asingle value in the theoretical limit. Regarding the average thermalconductivity, all simulation conditions result in a similar value,which again demonstrates the equivalence of time-averaging andensemble-averaging [20]. The results for silicon are similar to thosefor solid argon, except for the different HCACF and thermal conduc-tivity values, which arise primarily from the different atomicmasses and interatomic potentials of the two materials.

Wemove on to quantitatively understand the effects of ttotal andtcorre;UL on uncertainty of the EMD-predicted thermal conductivityof silicon. In Fig. 7(a), we show the variation of the average thermalconductivities with ttotal. For all tcorre;UL values, kave converges whenttotal is longer than around 10 ns. The converged kave generallyincreases with the increasing tcorre;UL, but changes negligibly aftertcorre;UL ¼ 400 ps, which confirms the appropriateness of the choiceof tcorre;UL for the results in Fig. 6. The much larger EMD-predictedthermal conductivities than the experimental value (80W/m-K)by Glassbrenner and Slack [21] could be attributed to the Tersoffpotential or the defects in the samples used in the experiments.In Fig. 7(b), we show the variation of the standard deviation ofthe thermal conductivities with ttotal, which also includes someresults for silicon at 1000 K. For all tcorre;UL values, rk decreases asttotal increases. We fit the data with a power law relation asrk � t�ccorre;UL, and the resulted c varied from 0.48 to 0.52. Similarto the solid argon results, we plotted rk=kave as a function ofttotal=tcorre;UL, as shown in Fig. 7(c). By fitting the data with a

square-root relation, we obtained rk=kave ¼ 2:006ðttotal=tcorre;ULÞ�0:5,which is in remarkable agreement with the result for solidargon.

3.3. Germanium

We also conducted EMD simulations of germanium with adomain size of 4� 4� 4 u.c. at 500 K. The results are similar to

Fig. 4. (a) Variation of the EMD-predicted average thermal conductivity of solidargon at 40 K with ttotal . The experimental value (0.560 W/m-K) by Touloukian et al.[19] is shown as a comparison. (b) Variation of the thermal conductivity of solidargon at 40 K with tcorre;UL. (c) Variation of the thermal conductivity of solid argon at40 K with ttotal . The insets in (b) and (c) show the histogram distributions of thethermal conductivities corresponding to the simulation conditions. The bell-shapedcurves represent normal distribution curves with the same average thermalconductivity and uncertainty.

Fig. 5. (a) Variation of the uncertainty of the EMD-predicted thermal conductivityof solid argon with tcorre;UL. (b) Variation of the uncertainty of the thermalconductivity of solid argon with ttotal . The results in (a) and (b) correspond to asolid argon material system with a domain size of 6� 6� 6 u.c. at 40 K. (c)Variation of the relative uncertainty of the thermal conductivity of solid argon withttotal=tcorre;UL corresponding to three domain sizes and two temperatures. Theformula in (c) represents a fit of the data with a square-root relation.

272 Z. Wang et al. / International Journal of Heat and Mass Transfer 112 (2017) 267–278

those for silicon. The fitted square-root relation turned out to be

rk=kave ¼ 1:887ðttotal=tcorre;ULÞ�0:5, which is in good agreement withthe results for solid argon and silicon. More details about the ger-manium results can be found in Supplemental Information.

3.4. Combined results and general consideration

Having examined the uncertainty of the EMD-predicted thermalconductivities of solid argon, silicon, and germanium, we providesome discussion on quantifying of the general uncertainty ofEMD-predicted thermal conductivities. In Fig. 8, we show the

Fig. 6. (a)–(d) Typical HCACF profiles of silicon with a domain size of 4� 4� 4 u.c. at 500 K. The simulations correspond to tcorre;UL ¼ 400 ps but different ttotal values. In (d) theresults based on single- and double-exponential fitting are included to show how the effective phonon relaxation time, seff , could be obtained. (e)–(h) Distributions of thermalconductivity integration profiles of silicon. The simulation conditions are indicated in the subfigures. The red (thin) curves represent the thermal conductivities from 100independent simulations (including a total of 300 curves). The blue (thick) curves represent the corresponding average thermal conductivities. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version of this article.)

Z. Wang et al. / International Journal of Heat and Mass Transfer 112 (2017) 267–278 273

variation of the relative uncertainty of the EMD-predicted thermalconductivities with ttotal=tcorre;UL, including the results for solidargon, silicon, and germanium under different simulation condi-tions. It is seen that the data for the different materials and simu-lation conditions follow a similar trend. When ttotal=tcorre;UL is small,the relative uncertainties are large and could be even larger than100%. As ttotal=tcorre;UL increases, the relative uncertainty decreases.Except for a few data points at extremely small or large ttotal=tcorre;ULvalues, all the data can be fit with a square-root relation, as

rk

kave¼ 2

ttotaltcorre;UL

� ��0:5

: ð4Þ

The independence of the square-root relation of the materialsystem or simulation condition suggests its wide applicability toother material systems or simulation conditions. In other words,this square-root relation could be universal and applicable to allEMD simulations. The previously obtained, different leading con-stants, 2.025, 2.006, and 1.887 for solid argon, silicon, and germa-nium, respectively, as compared with the ‘‘2” in the ‘‘universal”relation, could be attributed to the limited number of simulations,ttotal; tcorre;UL, and domain size. Note that all the simulation condi-tions included in Fig. 8 have sufficiently long tcorre;UL values, whichensure physically correct thermal conductivity predictions. Wepoint out that according to Section 2, the rk

kavein Eq. (4) should

essentially be understood as rkxkx;ave

. A derivation of Eq. (4) is available

in Supplemental Information.

3.5. How to choose tcorre;UL; ttotal, and N for EMD?

After understanding the uncertainty of EMD-predicted thermalconductivities, we consider a practical question. That is, howshould the tcorre;UL and ttotal for EMD simulations be appropriatelychosen? Because of the statistical nature of MD simulations,there is no easy answer to this question. Here we approachthis question by considering the relative error bound (Q) togetherwith the confidence level (P). We consider first the choice oftcorre;UL and then the choices of ttotal for N ¼ 1 and N � 2,respectively.

3.5.1. Choice of tcorre;ULFor EMD simulations to provide physically correct thermal con-

ductivity predictions, it is required that the tcorre;UL be sufficientlylong so that the truncation error is negligible. In real practice,tcorre;UL is usually determined by inspection of the HCACF profiles,which, although works in some cases, lacks consistency, becausedifferent researchers could have very different choices. Here weprovide a guideline for choosing tcorre;UL by considering an effectivephonon relaxation time, seff , which can be often obtained in twoways: (1) the single-exponential fitting of a HCACF, and (2) thedouble-exponential fitting of a HCACF.

The single-exponential fitting of a HCACF is based on theconcept of gray phonons, i.e., all phonons have the same effectiverelaxation time. The relevant formula reads,

Fig. 7. (a) Variation of the EMD-predicted average thermal conductivity of siliconwith ttotal corresponding to a domain size of 4� 4� 4 u.c. at 500 K. The experi-mental value (80 W/m-K) by Glassbrenner and Slack [21] is shown as a comparison.(b) Variation of the uncertainty of the thermal conductivity of silicon with ttotal . (c)Variation of the relative uncertainty of the thermal conductivity of silicon withttotal=tcorre;UL. The results in (b) and (c) correspond to a silicon material system with adomain size of 4� 4� 4 u.c. at 500 and 1000 K. The formula in (c) represents a fit ofthe data with a square-root relation.

Fig. 8. Variation of the relative uncertainty of the EMD-predicted thermal conduc-tivity with tcorre=tcorre;UL. The data for the different materials and simulationconditions follow a similar trend, which was fit with a single square-root relation.The numbers in the parentheses represent the average thermal conductivities fromthe corresponding EMD simulations. Note that all the simulation conditionsincluded in this figure have sufficiently long tcorre;UL values, which ensure physicallycorrect thermal conductivity predictions.

274 Z. Wang et al. / International Journal of Heat and Mass Transfer 112 (2017) 267–278

HCACF1ðtcorreÞ ¼ A1 exp � tcorreseff ; 1

� �; ð5Þ

where A1 and seff ; 1 are fitting parameters. Alternatively, A1 could befixed at HCACF(0), leaving seff ; 1 as the only fitting parameter. Butwe find that this latter method typically results in poorer fitting

results as compared to the first method. We used this method tofit the HCACF for solid argon at 40 K, silicon at 500 K, and germa-nium at 500 K, as seen in Fig. 3(d) for solid argon and Fig. 6(d) forsilicon (the germanium result is available in SupplementalInformation).

The double-exponential fitting of a HCACF is based on the for-mula, [5]

HCACF2ðtcorreÞ ¼ A21 exp � tcorreseff; 21

� �þ A22 exp � tcorre

seff; 22

� �; ð6Þ

where A21; seff; 21; A22, and seff; 22 are fitting parameters. Thismethod divides phonons into two broad categories: those with ashort relaxation time and those with a long relaxation time. Despitethis relatively coarse treatment, this method typically providesmuch better fitting results than the single-exponential fittingmethod. We used this method to fit the HCACF for solid argon at40 K, silicon at 500 K, and germanium at 500 K, as seen in Fig. 3(d) for solid argon and Fig. 6(d) for silicon (the germanium resultis available in Supplemental Information).

Table 1 summarizes the effective phonon relaxation timesobtained from the single- and double-exponential fitting. It is seenthat seff ; 1 lies between seff ; 21 and seff ; 22 (roughly at the middle). Asa validation, the solid argon results agree well with previousresults by McGaughey and Kaviany [5]. Considering the higheraccuracy of the double-exponential fitting, we recommend usingseff ; 22 (the longer time constant) as the effective phonon relaxationtime to determine the tcorre;UL. Hereafter, we will refer to this timeconstant as seff for simplicity. From Eq. (6), we can obtain the rel-ative truncation error due to a finite tcorre;UL, as

Etruncðtcorre;ULÞ ¼R1tcorre;UL

HCACF2ðtcorreÞ dtcorreR10 HCACF2ðtcorreÞ dtcorre

: ð7Þ

Substituting Eq. (6) into Eq. (7) and carrying out the integra-tions, we obtain

Etruncðtcorre;ULÞ ¼A21seff; 21 exp � tcorre;UL

seff ; 21

� �A21seff; 21 þ A22seff; 22

þA22seff ; 22 exp � tcorre;UL

seff ; 22

� �A21seff; 21 þ A22seff; 22

: ð8Þ

Fig. 9. Contour plot ttotal=tcorre;UL with the relative error bound (Q) and confidencelevel (P) for EMD simulations. This figure provides a guideline for choosing ttotal (thechoice of tcorre;UL is discussed in Section 3.5.1) to achieve a desired relative errorbound with a desired confidence level. The ttotal should be replaced by N � ttotal , if Nindependent simulations are conducted, and by 3N � ttotal , if the material can befurther assumed to be isotropic. The horizontal dotted lines indicate relative errorbounds of 10% and 20%. The vertical dotted lines indicate confidence levels of 68.3%and 90%.

Table 1Summary of effective phonon relaxation times for solid argon at 40 K, silicon at 500 K, and germanium at 500 K, obtained from the single- and double-exponential fitting.

Material Single exp. fit Double exp. fit

A1 (eV2A2/ps2) seff ; 1 (ps) A21 (eV2A2/ps2) seff; 21 (ps) A22 (eV2A2/ps2) seff ; 22 (ps)

Ar at 40 K 0.759 1.565 1.029 0.274 0.332 3.769Si at 500 K 811.7 16.65 568.3 4.650 450.9 28.45Ge at 500 K 369.8 18.54 246.7 6.290 195.2 31.75

Z. Wang et al. / International Journal of Heat and Mass Transfer 112 (2017) 267–278 275

Considering seff ; 21 � seff ; 22 and A21seff ; 21 � A22seff ; 22, which aretypically true, we have

Etruncðtcorre;ULÞ � exp � tcorre;ULseff; 22

� �: ð9Þ

As a result, we recommend choosing tcorre;UL to be ð5 � 10Þseff toachieve an Etrunc smaller than 1% (Note that expð�5Þ � 0:67%). Inthis study, most of the tcorre;UL values were conservatively chosento be larger than 10seff . In practice, we can first run a few EMD sim-ulations with a relatively long tcorre;UL (e.g., 100 ps), and then fit theaveraged HCACF profile (or the initial decaying portion of the aver-aged HCACF profile) from those EMD simulations with a double-exponential scheme to determine an appropriate tcorre;UL. Note thatthe double-exponential fitting only needs to be done once. Aftertcorre;UL is determined, the additional EMD simulations can be doneby using the direct integration method to calculate the thermalconductivity.

3.5.2. Choice of ttotal with N ¼ 1Here we consider a scenario, in which only one EMD simulation

is conducted to calculate the thermal conductivity. From theresults shown in Figs. 2 and 4, we realize that the thermal conduc-tivities from independent simulations can be reasonably assumedto follow a normal distribution. As a result, the predicted thermalconductivity in a particular direction (say kx, without loss of gener-ality) from just one simulation will fall in the range between(kx;ave � rkx ) and (kx;ave þ rkx ) with a confidence level of 68.3%.

If we define the relative error of an EMD-predicted thermal

conductivity as jkx;1�kx;ave jkx;ave

, where kx;1 is the thermal conductivity

from a single EMD simulation and kx;ave is the average thermalconductivity from a large number of EMD simulations (or the‘‘true” thermal conductivity), and the bound of the relative erroras Q, then the confidence level, P, corresponding to Q can beexpressed as

P ¼ Pjkx;1 � kx;avej

kx;ave6 Q

� �; ð10Þ

or,

P ¼ P kx;aveð1� QÞ 6 kx;1 6 kx;aveð1þ QÞ½ : ð11ÞConsidering the cumulative distribution function (CDF) of a nor-

mal distribution, Nðl;r2Þ (with a mean l and a standard devia-tion r) [22],

CDFðxÞ ¼ 12

1þ erfx� lrffiffiffi2

p� �� �

; ð12Þ

we have

P ¼ CDF kx;aveð1þ QÞ½ � CDF kx;aveð1� QÞ½ ; ð13Þwhich can be simplified by applying Eq. (12) and the property thaterfð�xÞ ¼ �erfðxÞ, into

P ¼ erfkx;aveQ

rkx

ffiffiffi2

p !

: ð14Þ

Finally, we can substitute Eq. (4) into Eq. (14) to obtain

P ¼ erf

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffittotal

8tcorre;UL

sQ

!; ð15Þ

which can alternatively be re-arranged as

ttotaltcorre;UL

¼ 8erf�1ðPÞ

Q

" #2: ð16Þ

Here the ‘‘erf and ‘‘erf�1” stand for the error function and inverseerror function, respectively. Therefore, the choice of ttotal directlydepends on the desired relative error bound and confidence level.

In Fig. 9, we show a contour plot of ttotaltcorre;UL

as a function of P and

Q. We focus on relative error bounds in the range from 0% to 50%and confidence levels from 0% to 100%. It is seen that at a constantQ ; ttotal

tcorre;ULincreases with the increasing P, whereas at a constant

P; ttotaltcorre;UL

decreases with the increasing Q. Note that only the results

with ttotaltcorre;UL

P 1 are physically correct, because ttotal P tcorre;UL. In the

following we provide two specific examples to illustrate the use ofEq. (16). (1) Consider kx of solid argon at 40 K, if we target atQ ¼ 10% and P ¼ 90%, and choose tcorre;UL ¼ 40 ps (around10:6seff ), then ttotal ¼ 43:3 ns. (2) Consider kx of silicon at 500 K, ifwe target at Q ¼ 20% and P ¼ 68:3%, and choose tcorre;UL ¼ 400 ps(around 14:1seff ), then ttotal ¼ 40:1 ns. We notice that the ttotal val-

276 Z. Wang et al. / International Journal of Heat and Mass Transfer 112 (2017) 267–278

ues are typically very large in order to achieve a relatively smallrelative error bound with a relatively high confidence level. Thelarge ttotal values, however, could be reduced by conducting multi-ple simulations, as discussed in the following section.

3.5.3. Choice of ttotal with N � 2It is a typical practice that EMD simulations are conducted with

multiple independent runs, and the average thermal conductivityis calculated. According to the Central Limit Theorem, [22] if Nindependent simulations are conducted, then the standard devia-tion of the average thermal conductivity distribution will decreasefrom rkx to

rkxffiffiffiN

p . To illustrate this fact, we consider the distribution of

the average thermal conductivity of solid argon calculated from Nindependent simulations (i.e., kx;ave;N), as shown in Fig. 10. The sim-ulations correspond to solid argon at 40 K with a domain size of6� 6� 6 u.c., tcorre;UL ¼ 40 ps, and ttotal ¼ 4:8 ns. We varied N from1 to 15. It is seen that the distributions corresponding to differentN values all qualitatively follow a normal distribution. We alsoobserve that the mean of the distribution remains the same(0.547W/m-K), but the standard deviation decreases, as Nincreases.

Substituting rkx with rkxffiffiffiN

p and repeating the derivations in Sec-

tion 3.5.2, we obtain

N � ttotaltcorre;UL

¼ 8erf�1ðPÞ

Q

" #2: ð17Þ

Consider again the two examples given in Section 3.5.2. Werealize that the ttotal for the first and second examples can bereduced to 4.33 and 4.01 ns, respectively, by conducting 10 inde-pendent simulations (i.e., N ¼ 10). Eq. (17) can also be used todetermine N, if ttotal is restricted upfront. It is worth mentioningthat Eq. (16) provides a mathematical demonstration of the equiv-alence of time-averaging and ensemble-averaging. At a fixedtcorre;UL, a desired Q and P can be achieved by maintaining(N � ttotal) to be a constant, which can be realized by running eithera small number of long simulations or a large number of shortsimulations.

Fig. 10. Distribution of the average thermal conductivity of solid argon from Nindependent simulations. The simulation conditions are indicated in the figure. Thedistributions corresponding to different N values all qualitatively follow a normaldistribution. The mean of the distribution remains the same (0.547 W/m-K), but thestandard deviation decreases, as N increases.

For isotropic materials, Eq. (17) can be further written as

3N � ttotaltcorre;UL

¼ 8erf�1ðPÞ

Q

" #2; ð18Þ

where the ‘‘3” accounts for the x; y, and z directions. Therefore, tak-ing into account the isotropicity of materials could further reducettotal by 66.7%.

We provide a validation for Eq. (18) by using the data for solidargon at 40 K with a domain size of 6� 6� 6 u.c., tcorre;UL ¼ 40 ps,ttotal ¼ 4:8 ns, and N ¼ 6, which are shown in Fig. 10. By consider-ing the combinations of 6 simulations out of 100 simulations(100C6 ¼ 1;192;052;400), we find that the confidence level for arelative error bound of 10% (i.e., 0:9kx;ave 6 kx;ave;6 6 1:1kx;ave, or0:492 6 kx;ave;6 6 0:602) is 98.5%. On the other hand, from Eq.

(18), we have P ¼ erfffiffiffiffiffiffiffiffiffiffiffiffiffi3�6�4:88�0:04

q� 10%

� �= 98.0%. The excellent

agreement of these two confidence levels provides a validationfor Eq. (18) (or the original Eq. (16)). The slight discrepancy couldbe attributed to the limited total number of simulations (i.e., 100).

3.6. How to report EMD-predicted kx?

Having understood the uncertainty of EMD simulations, weconsider how EMD-predicted thermal conductivities should bereported. In Fig. 11(a), we show a normal distribution with a mean(or average), kx;ave, and a standard deviation, rkx . If one simulationis conducted, then the kx will have a confidence level of 68.3% tofall between kx;ave � rkx and kx;ave þ rkx . We note that typically aconfidence level of 68.3% is assumed by default in the literaturewhen EMD-predicted thermal conductivities are reported witherror bars. In Fig. 11(b), we show three possible methods of report-ing EMD-predicted kx: (M1) using kx;ave;N and rkx ;N , (M2) using

kx;ave;N and rkx ;N=ffiffiffiffiN

p(named the ‘‘standard error”), and (M3) using

kx;ave;N and rkx ;N . In all the three methods, the average value of the NEMD simulations, kx;ave;N is considered the best estimate of the‘‘true” thermal conductivity, kx;ave, but they report the error barin different ways. In (M1), the error bar emphasizes the distribu-tion of the kx values from the N simulations (i.e., precision), insteadof how close the predicted kx;ave;N is to the kx;ave (i.e., accuracy). Alarger N could even counter-intuitively result in a larger errorbar. As a result, we consider this method to be inappropriate forreporting EMD-predicted kx, despite the fact that it has beenwidely used in previously studies. In (M2), the error bar empha-sizes the distribution of kx;ave;N (rather than kx) and thus the accu-racy of the predictions. It also correctly reflects the fact that alarger N will result in a smaller error bar. As a result, we considerthis method to be appropriate for reporting EMD-predicted kx. In(M3), the error bar is evaluated by using Eq. (4), where the ttotalshould be replaced by N � ttotal if N simulations are conducted. Sim-ilar to (M2), (M3) emphasizes the accuracy of the predictions. Weconsider this method to be appropriate for reporting EMD-predicted kx, because Eq. (4) is obtained from statistical analysisof a large number of data, and (M3) also incorporates the equiva-lence of time-averaging and assemble-averaging. Under theassumption that kx;ave;N � kx;ave (or Q � 0), (M2) and (M3) can beshown to be equivalent by using Eq. (4). To illustrate how thesethree methods work, we consider EMD-predicted kx of solid argonat 40 K with a domain size of 6� 6� 6 u.c. and tcorre;UL ¼ 40 ps, asshown in Fig. 11(c) and (d), which correspond to (ttotal ¼ 0:64 ns,N ¼ 100) and (ttotal ¼ 64 ns, N ¼ 1), respectively. From Fig. 11(c),it is seen that (M1) results in a large error bar, even though 100simulations are conducted, whereas (M2) and (M3) results in smalland nearly equally-sized error bars. From Fig. 11(d), it is seen thatno error bar is predicted by (M1) and (M2) since only one simula-

Fig. 11. (a) Illustration of a normal distribution with a mean (or average), kx;ave, and a standard deviation, rkx . (2) Illustration of three possible ways to report kx from EMDsimulations. (3) An example of reporting kx from 100 EMD simulations with a short ttotal . (4) An example of reporting kx from one EMD simulation with a long ttotal . The errorbars are not drawn to scale.

Z. Wang et al. / International Journal of Heat and Mass Transfer 112 (2017) 267–278 277

tion is conducted, but (M3) predicts an error bar, which is nearlythe same with the (M3)-predicted error bar in Fig. 11(c). Accordingto the equivalence of time-averaging and assemble-averaging, theerror bar for (ttotal ¼ 0:64 ns, N ¼ 100) and (ttotal ¼ 64 ns, N ¼ 1)should be same. Since (M3) correctly predicts this equivalence,we consider it to be the most appropriate method among the threefor reporting EMD-predicted kx. We recommend using (M3) forfuture EMD studies.

In the following we provide a summary of the key steps of doingEMD simulations.

(1) Choose a tcorre;UL according to the guideline in Section 3.5.1.(2) Choose a desired Q and P (typically 68.3%).(3) Choose a ttotal according to the guideline in Section 3.5.2, or a

ttotal and an N according to the guideline in Section 3.5.3.(4) Conduct the EMD simulations.(5) Report the EMD-predicted thermal conductivities according

to the guideline in Section 3.6.

A few remarks: (1) A ttotal or a ttotal and an N can also be deter-mined first, and then the Q calculated with Eqs. (16), (17), or (18)under a certain P (typically 68.3%). (2) The time step of the EMDsimulations should be chosen appropriately to capture the physicsof the phonon transport at an acceptable computational cost. In theliterature, it is typically chosen as around 1/50th of the period ofthe highest frequency phonon, which can be determined from lat-tice dynamics calculations, for example, by using GULP [23]. Weexpect that the time step could be chosen as large as around1/10th of the period of the highest frequency phonon; further stud-ies could be conducted on this topic. (3) A size effect study shouldbe conducted before meaningful thermal conductivity data arecollected.

Finally, we point out that although this study focuses onEMD-predicted thermal conductivities of isotropic materials, thekey findings (e.g., Eqs. (4) and (16)) should also apply to other

properties calculated with the Green-Kubo formula or its counter-part(s) (e.g., interfacial thermal conductance from EMD simulations[24]), or anisotropic materials (e.g., graphite, bismuth telluride). Inaddition, we have done some sensitivity analysis to evaluate therelative effects of the velocity initialization seed, tcorre;UL, and ttotalon the thermal conductivity predictions, as shown in SupplementalInformation.

4. Conclusions

In summary, this paper provides a study on quantifying theuncertainty of the EMD-predicted thermal conductivities by usingsolid argon, silicon, and germanium as model materials systems.We find that the uncertainty increases with the upper limit ofthe correlation time, tcorre;UL, and decreases with the total simula-tion time, ttotal, whereas the velocity initialization seed, simulationdomain size, temperature, and type of material have minimaleffects. We have obtained a ‘‘universal” square-root relation for

quantifying the uncertainty, as rkx=kx;ave ¼ 2ðttotal=tcorre;ULÞ�0:5. Withthis relation, it is possible to predict the uncertainty of the thermalconductivities from EMD simulations based on the chosen simula-tion parameters, even before the simulations are done. We havealso conducted statistical analysis of the EMD-predicted thermalconductivities and derived a formula that correlates the relativeerror bound (Q), confidence level (P), tcorre;UL; ttotal, and number ofindependent simulations (N). We recommend choosing tcorre;UL tobe 5–10 times the effective phonon relaxation time, seff , andchoosing ttotal and N based on the desired relative error boundand confidence level. We also recommend reportingEMD-predicted thermal conductivities as kx;ave;Nð1 QÞ, with theconfidence level indicated. This study provides new insights intounderstanding the uncertainty of EMD-predicted thermalconductivities. It also provides a guideline for running EMDsimulations to achieve a desired relative error bound with a

278 Z. Wang et al. / International Journal of Heat and Mass Transfer 112 (2017) 267–278

desired confidence level and for reporting EMD-predicted thermalconductivities.

Acknowledgments

Z. Wang and X. Ruan would like to thank the support from theNational Science Foundation (Award No. 1150948). S. Safarkhaniand G. Lin would like to acknowledge the support by the U.S.Department of Energy, Office of Science, Office of Advanced Scien-tific Computing Research, Applied Mathematics program as part ofthe Multifaceted Mathematics for Complex Energy Systems(M2ACS) project and part of the Collaboratory on Mathematicsfor Mesoscopic Modeling of Materials project, and NSF GrantDMS-1555072.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.ijheatmasstrans-fer.2017.04.077.

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